Implementation of gray image encryption with pixel shuffling and gray ...

Telecommun Syst DOI 10.1007/s11235-011-9461-0

Implementation of gray image encryption with pixel shuffling and gray-level encryption by single chaotic system C.K. Huang · C.W. Liao · S.L. Hsu · Y.C. Jeng

© Springer Science+Business Media, LLC 2011

Abstract When the grays of images present a strong contrast, merely encrypting the gray-level of image pixels is unable to eliminate the image outlines. The adoption of pixel shuffling can eliminate the image outlines, but cannot disrupt the characteristics of gray-level spectrum. Since a chaotic system is highly sensitive to the initial values and the chaotic trajectory is unpredictable, the application of this method to communication encryption and decryption renders a high level of security. In this paper, the chaotic system is adopted as the fundamental base and combined with row, column shuffling, and gray-level encryption to not only eliminate image outlines, but also disrupt the distributional characteristics of gray level, and the resulting increases of key space. Various statistical methods, such as correlation coefficient, NPCR (Number of Pixels Change Rate), UACI (Unified Average Changing Intensity), and entropy, provide analysis of the efficacy of the encryption method proposed. Our example reveals that the proposed encryption method can obtain highly secure encrypted images.

C.K. Huang () · C.W. Liao · Y.C. Jeng Department of Industrial Education and Technology, National Changhua University of Education, Changhua City, 500, Taiwan e-mail: [email protected] C.W. Liao e-mail: [email protected] Y.C. Jeng e-mail: [email protected] S.L. Hsu Department of Leisure Services, Chaoyang University of Technology, Wufong Township, Taichung County, 41349, Taiwan e-mail: [email protected]

Keywords Pixel shuffling · Gray-level encryption · Chaotic system · Correlation coefficient · Entropy

1 Introduction Data transmission has become an important focus for future development in the information industry. How to ensure information transmission security is a crucial issue. For decades, a many researchers and scholars have investigated the issue of data encryption and decryption in order to ensure security of communications during transmission. Since the dynamic response of the chaotic system is highly sensitive to the initial values and parameters of the method and the chaotic trajectory is unpredictable, its application to communication encryption and decryption is highly valued. To prevent data interception during transmission, the to-be-transmitted data is first hidden in chaotic status [1–3]. While few researchers have applied the scatter and disorder characteristics of the chaotic system to encryption and security communication, few have used these characteristics to the encryption of digital images. Although the application of a one-dimension chaotic method for image encryption is convenient and quick, it is not able to provide sufficient security [4]. Therefore, the use of a multiple dimension chaotic system is more appropriate in order to ensure communication security. In 1999, Lee and Chen [5] produced chaotic codes with the Lorenz chaotic system, applied these codes to the encryption and decryption of graylevel images, and obtained satisfactory results. Moreover, Yang and Yuan produced encryption codes out of chaotic sequences [6]. The fundamental principle was to take an amount of chaotic element as data sequence, transform them into binary codes, and subsequently shuffle the bits in order. Zhang and He combined chaotic sequences as encrypted

C.K. Huang et al.

Fig. 1 Flowchart of the encryption and decryption methods

matrices, encrypted the plain text matrix and then hid cipher text matrix with a two dimensional chaotic system for the sake of security enhancement [7]. In this paper, the chaotic system is the fundamental base. Usage of chaotic variable elements will be by row and by column shuffling coupled with gray-level value encryption on image pixels in order to enhance the security of encrypted images. Finally, statistical analysis tests image security.

2 Encryption and decryption Several scholars have researched image encryption conduct pixel-shuffling encryption on images by means of encryption algorithm [8]. Although this provides a certain degree of security, this encryption only eliminates the outlines of images. The method proposed in this paper of row, column, and gray-level encrypting, not only dissipates image outline, but also alters the distribution characteristics of gray-level values and accordingly enhances encryption security. Figure 1 shows the encryption and decryption scheme. 2.1 Encryption Abundant image statistic analysis proves that neighboring pixels in an image are closely related. On average, a high correlation exists between eight to sixteen pixels of the horizontal, vertical, and diagonal directions. To disturb this relationship and prevent interceptors from decoding, this paper uses chaotic variable sequences X, Y, Z, produced by chaos. Following is row and column chaotic shuffling on images with X and Y , and the gray encryption on the gray levels

of target images with Z in order to enhance the security of encrypted images. Figure 2 outlines the encryption process and the encryption procedures follow: Step 1. Input the initial values of the chaotic system, x0 , y0 , z0 . Step 2. Propose the gray-level matrix of original images AM×N . Step 3. Create chaotic variable element sequences X, Y , and Z, then elicit variable element matrices of the same dimension as original images A, X1(i,j ) , Y1(i,j ) . Step 4. Conduct column indexing and shuffling on A(i,j ) and X1(i,j ) one by one, with the function sortrows(•). The sortrows(•) are the function of sorting index, representing the ascending order of • sequence. Step 5. Judge whether j is equal to N , and if not, repeat step 4. Whenj = N is reached, the column shuffling of pixels is completed, and an encrypted image Ae1 is obtained. Figures 3 and 4 illustrate the index shuffling. Step 6. Conduct row indexing and shuffling on Ae1(i,j ) and Y1(i,j ) , one by one, with the function sortrows(•). Step 7. Judge whether i is equal to M, and if not, repeat step 6. When i = M is reached, the column shuffling of pixels is completed, and an encrypted image Ae2 is obtained. Figure 5 illustrates the index shuffling. Step 8. Transfer the encrypted image Ae2(M×N ) to Ae2(1×MN ) . Step 9. Conduct bits indexing and shuffling on Ae2(1×MN ) (binary) and Z, we can get the shuffled level matrix Ae2(1×MN ) .

Implementation of gray image encryption with pixel shuffling and gray-level encryption by single chaotic

Step 10. Conduct the dimension transform (1 × MN → M ×N ), and therefore the encrypted image Ae(M×N ) can be obtained. 2.2 Decryption Image decryption involves restoring the gray levels of the encrypted images, conducting row and column restoration based on the sequence matrix and the restored level matrix and finally obtaining a restored image, as shown in Fig. 6. The decryption procedures are: Step 1. Follow the steps 1∼3 of encryption to produce X, Y, Z, and X1(i,j ) , Y1(i,j ) of the same dimensions as Lena A. Step 2. Create a sequence matrix D0 of the same dimension as the matrix Ae2(1×MN ) (binary). Step 3. Conduct indexing and shuffling on D0 and Z with sortrows(•) and obtain the recovery matrix D0r . Step 4. Conduct bits indexing and restoring on Ae2(1×MN ) (binary) and D0r , we can get the restoration level matrix Ae2(1×MN ) . Step 5. Transfer the matrix Ae2(1×MN ) to Ae2(M×N ) . Step 6. Create a sequence matrix D of the same dimension as the original images, as shown in Fig. 7. Step 7. Conduct indexing and shuffling on D and Y1 with sortrows(•), and obtain the recovery matrix Dr of dimension M × N . Figure 8 shows the illustration. Step 8. Conduct row indexing and restoring on Ae2(i,j ) and Dr(i,j ) one by one, with sortrows(•).

Fig. 2 Procedures of row and column shuffling and gray level encrypting Fig. 3 Index shuffling of original Lena by map X1

Fig. 4 Illustration of column indexing and shuffling

C.K. Huang et al. Fig. 5 Illustration of row indexing and shuffling

Fig. 6 Flowchart of row and column restoring, and gray decryption

Step 9. Judge whether i is equal to M, and if i = M, repeat step 8. When i = M is reached, the row restoration is completed, and Ae1 is obtained. Figure 9 illustrates this indexing and restoration. Step 10. Create a sequence matrix D1 of the same dimension as the original image A, as shown in Fig. 10.

Step 11. Conduct indexing and shuffling on D1 and X1 with sortrows(•), and obtain the recovery matrix D1r . Figure 11 illustrates this indexing and shuffling. Step 12. Conduct column indexing and restoring on Ae1(i,j ) and D1r(i,j ) one by one, with sortrows(•). Step 13. Judge whether j is equal to N , and if j = N , repeat step 12. When j = N is reached, the original image


A is obtained. Figure 12 illustrates this indexing and restoring.

where α = 10, β = 14.78, γ = 0.0385, m0 = −1.27, m1 = −0.68. 3.1 Image encryption

3 Example

x˙ = α(y − x − h(x)),

(1a)

y˙ = x − y + z,

(1b)

z˙ = −βy − γ z,

(1c)

The first step is a dynamic simulation using the initial values of the chaotic system x0 = 0.1, y0 = 0.6, z0 = 0.1. Meanwhile, the function reshape(•) is used to produce the variable element sequence and matrices X1 , Y1 , Z. The function sortrows(•) is used to conduct column indexing and shuffling on the Lena images A and X1 and as a result, an encrypted image Ae1 is obtained, as shown in Fig. 13(c). Again, row indexing and shuffling on Ae1 and Y1 is conducted and an encrypted image Ae2 is obtained, as shown in Fig. 13(e). Finally, we performed gray levels encryption to obtain encrypted image Ae by Z. Figures 13(a)–(g) illustrates the image encryption method.

(1d)

3.2 Image decryption

The following example is presented in order to verify the plausibility of the proposed method, which encrypts and decrypts gray image Lena (256 × 256) as described in Sects. 3.1 and 3.2 and illustrate its efficacy. The following equation presents the Chua chaotic systems applied in this paper [9]:

h(x) = m1 x + 0.5(m0 − m1 )(|x + 1| − |x − 1|)

Fig. 7 Sequence matrix D

Fig. 8 Procedures of indexing and shuffling on D and Y1

Fig. 9 Illustration of indexing and restoring on Ae2 and Dr

The first step is the initial value inputting of the chaotic system x0 = 0.1, y0 = 0.6, z0 = 0.1 together with a dynamic simulation. Meanwhile, the function reshape(•) is used to produce variable element sequence and matrices (X1 , Y1 and Z). Following is decryption of the gray level of Ae by Dor (created by Z) occurs to obtain a restored image Ae2 , as shown in Fig. 14(c). The function sortrows(•) is again used to produce a sequence matrix D of the same dimension as the original image and conduct indexing and shuffling on D and Y1 to obtain a recovery matrix Dr . Once again, the function sortrows(•) is used to conduct row indexing and restoration on Dr and Ae2 , to obtain row restoration im-

C.K. Huang et al.

age Ae1 , as shown in Fig. 14(e). In addition, the function sortrows(•) is used to produce a sequence matrix D1 of the same dimension as the original image and conduct indexing and shuffling on D1 and X1 to obtain a recovery matrix D1r . Column indexing and restoration is performed on D1r and Ae1 , to obtain the restored Lena Ad , as shown in Fig. 14(g). Figures 14(a)–(g) shows the complete image decryption. 4 Security analysis This section describes analysis of the security of encrypted images with five statistical methods in order to measure encryption efficacy. 4.1 Key space

variation is x0 = 10−16 , y0 = 10−16 and z0 = 10−16 . Therefore, the key space of this encryption system can reach as high as 1048 . 4.2 Correlation coefficient Correlation coefficient measures the dependence of two adjacent variables at a certain direction. The more closely related these two variables are, the closer the correlation coefficient approaches 1. Conversely, if they are less closely related, the value of correlation coefficient approaches 0. The two variables are not related and unpredictable when the coefficient is close to 0. The calculation equation is [10]:

r=

The chaotic system adopted in this paper is highly sensitive to the trace variations of initial values and the sensitivity

n n i=1 Xi Yi )−( i=1 Xi )( i=1 Yi ) [n( ni=1 Xi2 )−( ni=1 Xi )2 ][n( ni=1 Yi2 )−( ni=1 Yi )2 ] n(

n

(2)

Table 1 Correlation coefficient tests Image

Axis Diagonal

Original Lena Encrypted Lena

Vertical

Horizontal

0.9198

0.9573

0.9201

−0.0025

−0.0006

−0.0050

Table 2 NPCR and UACI tests Image

Fig. 10 Sequence matrix D1

Fig. 11 Indexing and shuffling on D1 and X1

Fig. 12 Indexing and restoring of Ae1 and D1r

Encrypted Lena

Test method NPCR

UACI

99.54%

28.27%

Implementation of gray image encryption with pixel shuffling and gray-level encryption by single chaotic Fig. 13 A diagrammatic explanation of the image encryption: (a) original Lena, (b) index original Lena with X1 , (c) column-shuffled Lena Ae1 , (d) index Ae1 with Y1 , (e) rowand column-shuffled Lena Ae2 , (f) mapping Ae2 with Z, (g) encrypted Lena Ae

n where n( ni=1 Xi Yi ) − ( ni=1 i ) represents the nXi )( 2 i=1 Y sample and [n( i=1 Xi ) − ( ni=1 Xi )2 ] and n variation, [n( i=1 Yi2 ) − ( ni=1 Yi )2 ] are the sample standard variation of X and Y , respectively. Table 1 shows the outcome of the correlation coefficient calculation.

ensure that the pixels of every level matrix can be altered. UACI is, on the other hand, the percentage of the average level matrix change between the relational positions of two images. The following equations define NPCR and UACI [11]:

4.3 NPCR & UACI NPCR is the comparison of gray levels of relational positions between original and encrypted images in order to

m−1 n−1 NPCR =

i=0

j =0 D(i, j )

m×n

× 100%

(3)

C.K. Huang et al. Fig. 14 A diagrammatic explanation of the image decryption: (a) encrypted Lena, (b) mapping Ae with Dor , (c) gray level decrypted Lena, (d) index Ae2 with Dr , (e) rowrestored Lena Ae1 , (f) index Ae1 with D1r , (g) decrypted Lena

m−1 n−1 1 |A(i, j ) − ACS (i, j )| × 100% (4) UACI = m×n 255

4.4 Entropy

i=0 j =0

where D(i, j ) =

0, A(i, j ) = ACS (i, j ) , 1, A(i, j ) = ACS (i, j )

A is the original image of m × n dimensions, and ACS is the encrypted image. Table 2 lists the results of the NPCR and UACI calculations.

Rudolf Clausius (1864) was the first to propose the concept of entropy. This theory was first employed in information theory by Claude Elwood Shannon (1948). If the entropy of encrypted images is less than, but approaching eight, it will reduce the probability of successful restoration of images by interceptors. On the contrary, if the entropy is more than eight, then interceptors easily decode the encrypted images. The calculation of entropy is [12]:


entropy = −

N −1 2

Pi log2 Pi ,

(5)

i=0

where Pi is the probability of the pixel’s grey-level value i. After the calculation, the entropy of encrypted image Ae is 7.9967. 5 Conclusions This paper uses graphic method to demonstrate the main results for the image encryption, the application of variable elements of a chaotic system to pixel row and column shuffling and gray level encryption has resulted in highly secure images. Moreover, statistical analyses show that key space is 1048 , correlation coefficient of classical Lena encrypted images is less than 0.005 for all three directions, and entropy is 7.9967. In addition, NPCR and UACI prove it is possible to obtain excellent pixel alteration effects. Accordingly, the encryption technique proposed in this paper will render outstanding image encryption effects and high transmission security. References 1. Cuomo, K. M., Oppenheim, A. V., & Strogatz, S. H. (1993). Synchronization of Lorenz-based chaotic circuits with applications to communications. IEEE Transactions on Circuits and Systems II, Express Briefs, 40, 626–633. 2. Yang, T., & Chua, L. O. (1996). Secure communication via parameter modulation. IEEE Transactions on Circuits and Systems I, Fundamental Theory and Applications, 43, 817–819. 3. Yang, T., Wu, C. W., & Chua, L. O. (1997). Cryptography based on chaotic systems. IEEE Transactions on Circuits and Systems I, Fundamental Theory and Applications, 44, 469–472. 4. Kocarev, L. (2001). Chaos-based cryptography: a brief overview. IEEE Transactions on Circuits and Systems I, Fundamental Theory and Applications, 1, 6–21. 5. Lee, B. H., & Chen, Y. Y. (1999). Implementation of chaotic stream ciphers using chaotic pseudorandom number generator. Thesis of National Taiwan University, Taiwan. 6. Yang, W. A., & Yuan, D. M. (2006). A new method of key generation based on chaotic sequence. China Academic Journal Electronic Publishing House Mathematics in Practice and Theory, 36, 160–164. 7. Zhang, W., & He, R. C. (2006). An encryption and hiding algorithm based on logistic chaotic sequences. China Academic Journal Electronic Publishing House, Journal of Lanzhou Jiaotong University (Natural Sciences), 25, 80–82. 8. Sun, Z. J., Chen, Y., Wang, Y. X., & Liao, X. F. (2007). New image encryption algorithm based on Chua’s circuit. China Academic Journal Electronic Publishing House, Computer Engineering and Design, 28, 3328–3330. 9. Chua, L. O., & Lin, G. N. (1990). Canonical realization of Chua’s circuit family. IEEE Transactions on Circuits and Systems, 37, 885–902. 10. Bluman, A. G. (1998). Elementary statistics (3rd edn.). McGrawHill, New York. 11. Chen, G., Mao, Y., & Chui, C. K. (2004). A symmetric image encryption scheme based on 3D chaotic cat maps. Chaos, Solitons and Fractals, 21, 749–761.

12. Behnia, S., Akhshani, A., Ahadpour, S., Mahmodi, H., & Akhavan, A. (2007). A fast chaotic encryption scheme based on piecewise nonlinear chaotic maps. Physics Letters A, 366, 391–396. C.K. Huang received a Ph.D. degree in Engineering Science and Technology from National Yunlin University of Science and Technology, Taiwan, in 2004. He is now an associate professor with National Changhua University of Education. His current research involves chaotic system and image encryption.

C.W. Liao received both M.S. and Ph.D. in Industrial Education from National Taiwan Normal University, Taiwan, in 1994 and 2002, respectively. Since August 2007, he has been an associate professor with National Changhua University of Education in Taiwan. His research interests include technology and vocational education, energy education of technology, and image encryption.

S.L. Hsu received the Ph.D. degree in Department of Industrial Education & Technology, National Changhua University of Education, Taiwan. She also is an assistant professor in the Department of Leisure Services of Chaoyang University of Technology, Taiwan. Her research interests include human resources management, leisure services management, and image encryption.

Y.C. Jeng received his Ph.D. and Master degrees at Iowa State University. From 1988 to 2010, he is the professor at the Department of Industrial Education and Technology, National Changhua University of Education, Taiwan. His interesting research area is vocational & technical education, and his major teaching areas are engineering mechanics and vocational & technical education.